Four (algorithms) in one (bag): an integrative framework of knowledge for teaching the standard algorithms of the basic arithmetic operations

ABSTRACT

In this article we present an integrative framework of knowledge for teaching the standard algorithms of the four basic arithmetic operations. The framework is based on a mathematical analysis of the algorithms, a connectionist perspective on teaching mathematics and an analogy with previous frameworks of knowledge for teaching arithmetic operations with rational numbers. In order to evaluate the potential applicability of the framework to task design, it was used for the design of mathematical learning tasks for teachers. The article includes examples of the tasks, their theoretical analysis, and empirical evidence of the sensitivity of the tasks to variations in teachers’ knowledge of the subject. This evidence is based on a study of 46 primary school teachers. The article concludes with remarks on the applicability of the framework to research and practice, highlighting its potential to encourage teaching the four algorithms with an emphasis on conceptual understanding.

Keywords:

• Standard algorithms of the four basic arithmetic operations
• mathematical knowledge for teaching
• task design

Acknowledgement

This research was conducted at the Technion – Israel Institute of Technology. The Technion's generous financial help is gratefully acknowledged.

Notes

1. This article deals with the four standard algorithms being taught in Israel. The algorithms may vary in form in different countries.

2. A short version of this article was presented at the 8th Conference of the European Society for Research in Mathematics Education, Antalya, Turkey (Raveh & Koichu, 2013).

3. There are more mathematical concepts and laws that manifest themselves when particular algorithms are used. For example, some students’ mistakes can result from the violation of commutativity in the subtraction algorithm. The framework, however, is constructed as an integrative construct, and, accordingly, embraces only those concepts that underlie several algorithms.

4. The addition algorithm was not presented at the workshop because of time constraints.